Page 393 - Marks Calculation for Machine Design

P1: Naresh
January 4, 2005
15:28
Brown.cls
Brown˙C09
U.S. Customary MACHINE ENERGY SI/Metric 375
solution solution
Step 1. Using Eq. (9.22), determine the spring Step 1. Using Eq. (9.22), determine the spring
rate (k) as rate (k) as
F s 50 lb F s 225 N
k = = = 40 lb/in k = = = 7500 N/m
y 1.25 in y 0.03 m
Step 2. Substitute the spring rate (k) found Step 2. Substitute the spring rate (k) found
in step 1 and the other given information in in step 1 and the other given information in
Eq. (9.25) to determine the number of active Eq. (9.25) to determine the number of active
coils (N a ) as coils (N a ) as
4
4
d G d G
N a = N a =
3
3
8D k 8D k
2
6
4
2
9
4
(0.11 in) (11.5 × 10 lb/in ) (0.0027 m) (80 × 10 N/m )
N a = N a =
3
3
(8)(1in) (40 lb/in) (8)(0.025 m) (7500 N/m)
1,684 4.25
= = 5.26 → 6 coils = = 4.53 → 5 coils
320 0.9375
9.2.3 Work and Energy
Figure 9.3 can be used to provide an expression for the work done on or by a spring, or the
energy absorbed or released by a spring. If a linear spring is compressed or lengthened by
a displacement (x 1 ), then the area under the shaded triangle in Fig. 9.4 gives the work done
on or by the spring, or the energy stored or released by the spring.
F s F = kx
s
kx 1
k
kx 1
0 x
0 x 1
FIGURE 9.4 Work done or energy stored by a spring.
The area of the shaded triangle, denoted as (Work), is given in Eq. (9.26) as
1→2
1 1 1 2
Work = (base)(height) = (x 1 )(kx 1 ) = kx 1 (9.26)
1→2 2 2 2
where the displacement (x 1 ) is the difference between the final length and the unstretched
length. Units on (Work) are (ft · lb) in the U.S. Customary and (N · m) in the SI/metric.
1→2
The underscript (1→2) on Work in Eq. (9.26) represents the fact that work is done on
the spring from one position to another, meaning it is path dependent. In contrast, energy
is related to a specific position, regardless of the path to get to this position.